Pattern forming pulled fronts: bounds and universal convergence
Ebert, U.; Spruijt, W.; Saarloos, W. vanCitation
Ebert, U., Spruijt, W., & Saarloos, W. van. (2004). Pattern forming pulled fronts: bounds and universal convergence. Physica D: Nonlinear Phenomena, 199, 13-32.
doi:10.1016/j.physd.2004.08.001
Version: Not Applicable (or Unknown)
License: Leiden University Non-exclusive license Downloaded from: https://hdl.handle.net/1887/68177
arXiv:nlin/0312010v1 [nlin.PS] 5 Dec 2003
Pattern forming pulled fronts:
bounds and universal convergence
Ute Ebert
1,2, Willem Spruijt
3and Wim van Saarloos
31 CWI, Postbus 94079, 1090 GB Amsterdam, The Netherlands, 2 Department of Physics, TU Eindhoven, Postbus 513, 5600 MB Eindhoven,
The Netherlands,
3 Instituut–Lorentz, Leiden University, Postbus 9506, 2300 RA Leiden,
The Netherlands
Abstract
We analyze the dynamics of pattern forming fronts which propagate into an unstable state, and whose dynamics is of the pulled type, so that their asymptotic speed is equal to the linear spreading speed v∗. We discuss a method that allows to derive
bounds on the front velocity, and which hence can be used to prove for, among others, the Swift-Hohenberg equation, the Extended Fisher-Kolmogorov equation and the cubic Complex Ginzburg-Landau equation, that the dynamically relevant fronts are of the pulled type. In addition, we generalize the derivation of the universal power law convergence of the dynamics of uniformly translating pulled fronts to both coherent and incoherent pattern forming fronts. The analysis is based on a matching analysis of the dynamics in the leading edge of the front, to the behavior imposed by the nonlinear region behind it. Numerical simulations of fronts in the Swift-Hohenberg equation are in full accord with our analytical predictions.
1 Introduction
In the last few years, it has become clear that when considering a problem of a front which propagates into an unstable state, it is crucial to distinguish two different classes, according to whether their asymptotic speed is equal to or larger than the linear spreading speed v∗. The linear spreading speed is a
can be determined explicitly from a long-time asymptotic saddle-point type analysis of the Green’s function of the relevant dynamical equation. In prac-tice, therefore, v∗is given explicitly by the dispersion relation of Fourier modes
obeying the linearized dynamical equation [1,2,3,4,5,6,7].
Given the existence of a finite linear spreading speed v∗ for a given problem,
only two different types of asymptotic front solutions can emerge starting from “steep” or “sufficiently localized” initial conditions: either the asymptotic ve-locity of the nonlinear front is equal to v∗or it is larger than v∗. In the first case
150 200 250 300 350 400
x
-1 -0.5 0 0.5 1 uFig. 1. Snapshot of a front in the Swift-Hohenberg equation (10) for ε = 0.5. The front propagates to the right into the region where u is in the unstable state u = 0.
One of the simplest examples of a dynamical equation whose pattern forming fronts are coherent is the Swift-Hohenberg equation, and we will therefore use this equation to illustrate and test our analytical results. In fact, the Swift-Hohenberg equation has often played a role in studies of front propagation [17,18,19,20,21,22] — it is essentially the only equation with pattern forming fronts for which a number of exact results (including the convergence to a pulled front solution) are known [19,20,21,22].
Because there are so few rigorous results for pattern forming fronts in gen-eral, we will, before turning to the analysis of the front convergence, discuss a method which allows us to derive a bound on the velocity for pattern forming fonts, like the Swift-Hohenberg equation, the Extended Fisher-Kolmogorov equation, or the cubic Complex Ginzburg-Landau equation. Although our ar-gument is in essence a simplified version of the line of analysis Collet and Eckmann [22] use to prove that fronts in the Swift-Hohenberg equation are pulled, we do want to show the reader how in just a few lines one can prove that fronts in pattern forming equations are pulled: we think that the method holds the promise for many new rigorous results on front propagation.
2 The linear spreading velocity as a rigorous upper bound
2.1 The linear spreading velocity
We consider a generic dynamical equation for some generic dynamical vari-able φ, whose stationary state φ = 0 is linearly unstvari-able, and whose dispersion relation is given by ω(k). This means that a Fourier perturbation eikx of the
unstable state evolves under the linear dynamics as e−iω(k)t+ikx. Associated
with the linear dynamical problem is a linear spreading velocity v∗, the
ve-locity with which an initially localized perturbation spreads asymptotically into the unstable state according to the linearized dynamics. The asymptotic spreading is simply determined by a long-time saddle point analysis of the Green’s function of the linear equation. From this analysis, one finds v∗
ex-plicitly in terms of ω(k) as [1,2,3,4,5,6,7] dω(k) dk k∗ = Im ω(k∗) λ∗ , v ∗ = Im ω(k∗) λ∗ , k ∗ ≡ q∗+ iλ∗. (1)
The first equation determines the saddle point value k∗ in the complex plane,
and the second one then gives the linear spreading velocity v∗. The third
equa-tion fixes our notaequa-tion for the splitting of k∗ into real part q∗ and imaginary
part λ∗ for the remainder of the paper. The complex parameter
D ≡ i 2 d2ω(k) dk2 k∗ (2)
plays the role of a complex diffusion coefficient1. If there are several saddle
points, the one with the largest v∗ is the relevant one [6]. Since the growth
rate Im (−iω(k) + iv∗k) in the comoving frame is maximal at k∗ for a relevant
saddle point, the sign of ReD is fixed:
Re D > 0. (3)
Our analysis applies to sufficiently steep initial conditions [6] lim
x→∞φ(x, 0) e λx
= 0 for some λ > λ∗; (4)
initial conditions with bounded support fall into this class. An important result is that in a frame ξ = x− v∗t moving with velocity v∗ to the right, the 1 D is the complex generalization of the diffusion constant D as in [7] and should
asymptotic evolution of the field under the linear equation is given by φ(x, t)∼ e−λ∗ξ + iq∗ξ− iΩ(k∗)t e−ξ
2/4Dt
√
4πDt , (5)
where
Ω(k)≡ ω(k) − v∗k, (6)
and where the co-moving coordinate
ξ = x− v∗t (7)
is held fixed while t→ ∞. This follows from the saddle point analysis of the Green’s function in the limit of large t, cf. sections 5.3 and 5.5.1 in [6]. The saddle point equations (1) can be expressed in terms of Ω(k) as dkΩ|k∗ = 0 and Im Ω(k∗) = 0. For the remaining real part of Ω(k∗), we use the notation
Ω∗ = Ω(k∗) = Re Ω(k∗). (8)
Eq. (5) illustrates that an initially sufficiently localized linear perturbation reaches the velocity v∗ and the spatial decay rate λ∗ for t → ∞ under the
dynamics of the linearized equation.
2.2 Upper bounds on the velocity: proof of pulling
When a front evolves under the full nonlinear equation into an unstable state, its asymptotic speed can never be smaller than the linear spreading velocity v∗.
If the asymptotic speed equals v∗, the front is called pulled [5,6,7,8], otherwise
it is called pushed. As a rule of thumb, dynamical equations whose nonlinear terms are all suppressing the growth lead to pulled fronts, but there is at present no general theory that allows one to predict when fronts are pulled and when they are pushed.
In the present section, a simple proof is given that fronts in some pattern forming equations are pulled.
2.2.1 A real field φ with nonlinearityN (φ) φ
∂tφ = N
X
n=0
an∂xnφ− N (φ) φ, N (0) = 0, (9)
with explicit linear terms and a nonlinearity2 N (φ) φ. Examples of such
equations are the nonlinear diffusion equation ∂tu = ∂x2u + f (u), the
Swift-Hohenberg-equation
∂tu = εu− (∂x2+ 1)2u− u3 = (ε− 1)u − 2∂x2u− ∂x4u− u3, (ε > 0), (10)
or the Extended Fisher-Kolmogorov (EFK) equation [7,23,24]
∂tu = ∂x2u− γ∂x4u + u− u3. (11)
The linear operator determines the dispersion relation ω(k) and the parame-ters v∗, Ω∗, q∗, λ∗ and D as discussed above.
The relevant dynamics of a pulled front that leaves a homogeneous state be-hind (Ω∗ = 0 = q∗), was identified in [6] by the leading edge transformation
φ(x, t) = e−λ∗ξ
ψ(ξ, t), ξ = x−v∗t. For pattern forming fronts with Ω∗ 6= 0 6= q∗,
different generalizations of this transformation are possible. While in the next section dealing with the asymptotic dynamics, the complete complex phase e−λ∗ξ+iq∗ξ−iΩ∗t
will be factored out of φ, for deriving bounds, it will be more convenient here to factor out the envelope e−λ∗ξ
. In a frame moving with velocity v∗, the field ˆψ(ξ, t) is then defined through
φ(x, t) = e−λ∗ξψ(ξ, t),ˆ ξ = x− v∗t. (12)
The effect of the transformation is demonstrated by comparing Fig. 1 with Fig. 2 below which show the original dynamical field u of the Swift-Hohenberg equation and the associated field ˆψ. The field ˆψ in Fig. 2 magnifies the relevant dynamics in the leading edge which we will analyze in section 3, while this dynamics is hidden in Fig. 1.
With this transformation, the equation of motion for ˆψ becomes
∂tψˆ− v∗(∂ξ− λ∗) ˆψ = N X n=0 an(∂ξ− λ∗)nψˆ− N ˆ ψ e−λ∗ξ ψ.ˆ (13) With the two auxiliary functions of the Fourier variable k
2 Note that in [6] the complete nonlinear expressionN (φ) φ was denoted as N(φ),
σ(k) =
N
X
n=0
an(ik− λ∗)n+ v∗(ik− λ∗) =−iω(k + iλ∗) + iv∗(k + iλ∗),
¯ ψ(k, t) = ∞ Z −∞ dξ ˆψ(ξ, t) e−ikξ, (14)
the linear operators in Eq. (13) can be written in a more compact form
∂tψ(ξ, t) =ˆ ∞ Z −∞ dk 2π e ikξ σ(k) ¯ψ(k, t) − Nψ eˆ −λ∗ξ ψ.ˆ (15)
Now multiply the equation with ˆψ(ξ, t) and integrate over space. Using the identity Z dk σ(k) ¯ψ(k, t) ¯ψ(−k, t) =Z dk Re σ(k) ψ(k, t)¯ 2 , (16)
the final result is ∂ ∂t Z dξ ψˆ 2(ξ, t) 2 = Z dk 2π Re σ(k) ψ(k, t)¯ 2 −Z dξ N ψ(ξ, t) eˆ −λ∗ξ ψˆ2(ξ, t). (17) If φ initially is sufficiently steep (4) for x → ∞, and if |φ| stays bounded behind the front at x→ −∞, then the integrals exist initially. If furthermore the right hand side of (17) can be shown to be negative and of order R
dξ ˆψ2,
then R
dξ ˆψ2(ξ, t)↓ 0 for growing t. This means that in a frame moving with
velocity v∗, ˆψ2 vanishes; and this implies that the front cannot move faster
than v∗ for t→ ∞.
For the r.h.s. of (17) to be negative, we need both integrals to be negative. Since Re σ(k) = Im ω(k + iλ∗)− v∗λ∗, the saddle point construction entails
that Re σ(q∗) = 0, ∂
kσ|q∗ = 0 and ∂k2σ|q∗ =−2D with Re D > 0. Therefore
Re σ(k)≤ 0 for all real k. (18)
If there are several saddle points, this condition holds for the one corresponding to the largest spreading speed v∗ [6]. The present formulation in terms of σ(k)
yields another route to this conclusion.
The sign of the integral over the nonlinearity is fixed if the sign ofN is fixed. Therefore a sufficient criterion for the front to be pulled is
In a pattern forming front, the sign of φ can change. This increases the rele-vant values of φ and therefore decreases the admissible functions N . E.g., for N (φ) = φr, a monotonic front with non-negative φ will be certainly pulled
for all r > 0, while for a pattern forming front, r needs to be an even inte-ger. Both in the Swift-Hohenberg and EFK equation, N is quadratic in the dynamical variable, hence the above argument immediately shows that suffi-ciently steep initial conditions lead to pulled fronts in these equations. With a few slight modifications, the analysis can also be extended to the difference equation dCi/dt = Ci − Ci−12 , for which fronts were empirically found to be
pulled [14,25].
2.2.2 A complex field A: the complex Ginzburg-Landau-equation
It was already remarked by Collet and Eckmann in a footnote in [22] that the above line of analysis can be extended to the case of the cubic Complex Ginzburg Landau equation. We present the argument here in our language, and then generalize it to an even more general class of equations in the next subsection.
We analyze the complex Ginzburg-Landau-equation for complex field A(x, t) ∂tA = ǫA + (1 + c1)∂x2A− (1 − ic3)|A|2A, with ǫ, c1, c3 real, (20)
or more generally an equation of the form ∂tA =
N
X
n=0
an∂xnA− N (A) A, with A(x, t), an complex. (21)
The saddle point parameters λ∗, q∗, v∗, Ω∗ and D are again used for the
transformation ˆ
ψ(ξ, t) = e−λ∗ξ A(x, t), where ξ = x− v∗t. (22)
The calculation now follows essentially the lines of the previous calculation — except that one has to take into account that the field ˆψ and the coefficients are now complex. Therefore the equations of motion for A∗ or ˆψ∗ have to be
considered, too. They are, of course, derived by simply taking the complex conjugate of the equations for A and ˆψ. One then easily derives an equation for ˆψ∗∂
tψ + + ˆˆ ψ∂tψˆ∗ = ∂t| ˆψ|2 that after spatial integration and a few steps of
Here ¯ψ(k, t) and σ(k) are defined precisely as in (14).
This means that the complex equation has been reduced to expressions that contain absolute values and real parts only. Therefore the conclusion from the previous subsection is easily extended: an equation of form (20) or (21) creates pulled fronts if
Re N (A) ≥ 0 for all relevant A. (24)
This is a nontrivial result, since in contrast to the real equation (12), the com-plex equation does not have an energy minimizing structure; still the bound can be derived in the same way as before. Specialized to the cubic Complex Ginzburg-Landau equation, the above analysis simply proves that fronts in this equation are pulled, a fact known already empirically since over 20 years [7,26].
2.2.3 Generalization of admissible linearities and nonlinearities
In the last step, the admissible linear and nonlinear operators are reconsidered and generalized. For complex functions A, the general form is
LA + N (A, ∂xA, ∂x2A, . . . , ∂xmA) A = 0, (25)
where N again can be complex. L is an arbitrary complex linear operator that can take the differential form above, but also a difference or integral or mixed form as discussed in Section V of [6]. It determines the saddle point parameters v∗, λ∗, q∗ and D. Independent of the original functional form of
the linear operator, the expansion about the (large-t, large-x)-saddle point will lead to the differential form
τ0∂tψ = . . .ˆ − N (A, ∂xA, ∂x2A, . . . , ∂xmA) ˆψ. (26)
The analysis now proceeds as before with the final result ∂ ∂t Z dξ | ˆψ(ξ, t)| 2 2 = . . .− Z dξ Re N (A, ∂xA, . . . , ∂ m x A) τ0 | ˆ ψ|2. (27)
A sufficient criterion for the front to be pulled is Re N (A, ∂xA, . . . , ∂
m x A)
τ0 ≥ 0 for all relevant A.
In essence, the method discussed here confirms mathematically what one would expect intuitively for equations where only the linear terms lead to growth away from the unstable state φ = 0, while all the nonlinear terms are clearly stabilizing. In such cases, fronts are shown to be of the pulled type. There are several cases where fronts are empirically known to be pulled, but where the method in its present formulation fails. E.g., while adding a nonlinearity like −(∂xu)2u to the Swift-Hohenberg equation (10) or EFK
equation (11) leaves the fronts in these equations of the pulled type, since N = (∂xu)2 ≥ 0, the nonlinearity of the Kuramoto-Sivashinsky-equation
∂tu = −∂x2u − ∂4xu + (∂xu)u does not fall into the class (28). In fact,
ex-tending the method to the Kuramoto-Sivashinsky equation must clearly be quite a challenge, since adding a linear term c∂3
xu gives a transition to pushed
fronts for c≈ 0.15 [7]. An easier challenge to start with appears to be the the Cahn-Hilliard-equation ∂tu = −∂x2(∂x2u + u− u3). Again, in its present form
our method does not apply straightforwardly to the Cahn-Hilliard equation. Nevertheless, for a front penetrating the state u = 0 under the Cahn-Hilliard dynamics, we derive after a few partial integrations that
∂t Z dξ ˆψ2 = . . .− 3 Z dξ ˆψ2(∂xu)2− (λ∗u)2 . (29)
It is very likely that the sign of this integral over the nonlinearity is negative, since (∂xu)/u is the local slope of the full oscillating front, while (λ∗u)/u is
the slope of only the envelope in the leading edge. However, we have not yet been able to prove this.
In summary, we have derived sufficient criteria for a large class of equations to form pulled fronts, i.e., fronts that propagate with the linear spreading speed v∗. We now proceed to determining their actual rate of convergence to the
asymptotic behavior.
3 Power law convergence to the asymptotic speed and shape of a
pulled front
In [6] we have analyzed pulled fronts that for long times approach uniformly translating fronts, and we have derived their rate of convergence to the asymp-totic velocity and front profile. We will now extend this analysis to pattern forming fronts.
out order by order. It is remarkable and in line with the picture that has emerged for the pulled front mechanism, that the coefficients in the asymptotic expressions are actually obtained from the asymptotic analysis in the leading edge only; more precisely they are given by the saddle point parameters (1), (2) of the linearized equation. This is because for the analysis in the leading edge only input on the dominant analytic behavior of the asymptotic front profile is needed3. For brevity, we will therefore present here only the generalization of
the asymptotic expansion in the leading edge, following the lines of our earlier paper.
3.1 The dynamical equation for the leading edge variable ψ in the frame ξX
The first ingredient of the asymptotic analysis for the front convergence is to note that in the leading edge, the saddle point analysis from Section 2.1 implies that the field ψ(ξ, t) defined through
φ(x, t) = e−λ∗ξ eiq∗ξ−iΩ∗tψ(ξ, t), ξ = x− v∗t. (30)
becomes a function which varies slowly in space and time for large x and t, and this slow dynamics is governed by a generalized diffusion equation of the form ∂ψ ∂t =D ∂2ψ ∂ξ2 +D3 ∂3ψ ∂ξ3 + w ∂2ψ ∂t∂ξ + τ2 ∂2ψ ∂t2 +· · · − N (φ, . . .) ψ. (31)
In the function ψ, the full complex prefactor is factorized out of φ, in contrast to the partial factorization in Eq. (13). The parameter D is the generalized diffusion coefficient defined already in Eq. (2) above. Likewise the other ex-pansion coefficients D3, w, τ2 et cetera can all be expressed in terms of the
expansion of the dispersion relation near the saddle point — see Eq. (5.64) of [6]. E.g., we simply have D3 = (1/3!)d3ω/dk3|k∗. Note that we call Eq. (31) a generalized diffusion equation since the dominant terms for large ξ and t are in fact diffusive and can generate the Gaussian from Eq. (5).
For equations which lead to uniformly translating fronts, q∗ = 0 andD is real,
but for pattern forming fronts D is generally complex and q∗ 6= 0.
As discussed in [5,6,7], if we follow a level line where|φ| is constant, the 1/√t term in (5) implies an unbounded logarithmic shift in the position of the level line, and hence of the transient fronts in the nonlinear equation. The crux of
3 In the language of a matching analysis, the outer (leading edge) expansion of the
the convergence analysis is therefore to introduce a collective coordinate X(t) for the front position,
˙ X(t) = c1 t + c3/2 t3/2 + c2 t2 +· · · ⇐⇒ X(t) = c1ln t− 2c3/2 t1/2 +· · · , (32)
and to perform an expansion in the logarithmically shifted frame
ξX = ξ− X(t) = x − v∗t− X(t). (33)
For pattern forming fronts, we likewise introduce a global time-dependent phase Γ(t), ˙Γ(t) = d1 t + d3/2 t3/2 + d2 t2 +· · · ⇐⇒ Γ(t) = d1ln t− 2d3/2 t1/2 +· · · , (34)
and we define the field ψX in the shifted frame ξX and with a global slow
phase factor Γ by writing φ as φ(x, t) = e−λ∗ξx
eiq∗ξX−i(Ω∗t+Γ(t))
ψX(ξX, t). (35)
Comparison of (30) and (35) shows that
ψ(ξ, t) = eλ∗X(t)−iq∗X(t)−iΓ(t) ψX(ξX, t). (36)
With this transformation, we obtain from (31) the relevant dynamical equa-tion4 for ψ X(ξX, t) ∂ψX ∂t − ˙X(t) ik ∗+ ∂ ∂ξX ! ψX − i ˙Γ(t)ψX =D ∂2ψ X ∂ξ2 X +D3 ∂3ψ X ∂ξ3 X + . . . +w " ∂ ∂t − ˙X(t) ik ∗ + ∂ ∂ξX ! − i ˙Γ(t) # ∂ψX ∂ξX +· · · − N ψX. (37)
4 The term proportional to w is not present for equations like the Swift-Hohenberg
3.2 The asymptotic expansion for ψX in terms of similarity variables of the
diffusion equation
As we already pointed out above, in dominant order, the dynamical equation (31) for ψ(ξ, t) is a diffusion equation, and this was reflected by the fact that in the fully linear spreading problem, ψ(ξ, t) is just the fundamental Gaussian similarity solution e−ξ2
/(4Dt)/√t — Cf. Eq. (5). As explained in [6,16], the
nonlinearity in (31) can be interpreted as a sink for the diffusive field ψX
to the left of the leading edge. This imposes that in contrast to the linear problem, ψ has to increase linearly in ξ for small ξ. The relevant fundamental solution of the diffusion equation which has this behavior is
ψ(ξ, t)∼ ξ t3/2e
−ξ2
/(4Dt), (38)
and as explained in detail in [6,7] one can already obtain the dominant term of the power law relaxation of the velocity and front shape from this argument. The expansion is systematized by working in the ξX frame, as explained above,
and by recognizing that the similarity variable of the diffusion equation is
z = ξ
2 X
4Dt. (39)
In short, since far ahead of the front in the leading edge, ψX will fall off like
a Gaussian e−z = e−ξ2
/(4Dt) for a sufficiently steep front (4) (see [6]), we write
ψ(ξX, t) = G(z, t)e−z. (40)
To ensure the Gaussian decay for large ξX and finite t, we require
lim
z→∞G(z, t) e
−z = 0 ⇐⇒ lim ξx→∞
ψX(ξX, t) = 0. (41)
Note that as we already stated in (3), ReD > 0, so the limit z → ∞ should be taken along a line in the right complex z plane. This is the first boundary condition for G. The second boundary or matching condition arises from the behavior for small ξX, actually in the transition towards the nonlinear regime.
In agreement with the intuitive argument about the nonlinearity as a sink for the diffusion process, one derives
ψX(ξX, t)
ξX/√t→0
where α and β are in general complex constants with α 6= 0 due to the nonlinearity5.
Upon substitution of (40) into Eq. (37) for ψX, and using the expansion (32)
for X(t) and (34) for Γ(t), we obtain the equation of motion for G
t∂tG− c1+ c3/2 √ t ! " ik∗+ √ z √ Dt(∂z − 1) # G− i d1 + d3/2 √ t ! G = z∂2 z + 1 2 − z ∂z− 1 2 G +D3 √ z D32√t 3 2(∂z− 1) 2 + z (∂z− 1)3 G (43) +w √ z √ Dt h t∂t− z(∂z− 1) − 1 − ik∗c1− id1 i (∂z− 1) G + · · · .
The relevant long-time asymptotics of ψX then directly follows from solving
this equation with boundary conditions (41) and (42) [6]. As in [6], the coeffi-cients ci and di in X(t) and Γ(t) can be obtained by expanding G(z, t) as an
asymptotic series in terms of functions of the similarity variable z, G(z, t) = t1/2g−1/2(z) + g0(z) +
g1/2(z)
t1/2 +
g1(z)
t · · · , (t≫ 1), (44)
where the matching condition (42) implies that the leading order indeed is√t with the coefficient g−1/2(z) =√z + . . . for small z.
From here on, the analysis is just the technical implication of the expansion introduced above. Since the structure of the analysis follows essentially the one given in our earlier work on uniformly translating fronts, we relegate the details to appendix A. The final outcome of the analysis is that the velocity relaxes to v∗ according to the general formula
v(t)≡ v∗+ ˙X(t) = v∗− 3 2λ∗t + 3√π 2(λ∗)2t3/2 Re 1 √ D +O 1 t2 , (45)
5 For the nonlinear diffusion equation, we derived Dα = R∞
−∞dξ N ψ in section
2.5.2 of [6]. The relation between non-vanishing α and N can be generalized to pattern forming fronts [16]. In general, N then becomes time dependent and some temporal averaging is required. For the cubic CGL equation (20), however, we obtain Dα =R∞
−∞dξ (1+ic3)|A|2ψ(ξ) without temporal averaging. The phase of α changes
while the phase relaxation is governed by a similar expression, ˙Γ(t) = −q∗X(t)˙ − 3√π 2λ∗t3/2 Im 1 √ D +O 1 t2 . (46)
3.3 Convergence of a coherent front profile to its asymptotic shape
The above expressions are valid for any pulled front, irrespective of whether it is asymptotically uniformly translating or a coherent or incoherent pat-tern forming front6. Here ‘coherent’ means that the approximately periodic
pattern laid down by the leading edge of the front stays periodic in the non-linear region, while incoherent means that the pattern undergoes some further dynamics behind the front. Such incoherent fronts arise e.g. in some param-eter regimes of the cubic and quintic Complex Ginzburg Landau equation [7,16,26,27] or the Kuramoto-Sivashinsky equation [7]. Even when a pulled pattern forming front is incoherent the linear dynamics in the leading edge is described by the above equations. The dynamics in the leading edge is there-fore still coherent: the incoherent behavior only sets in in the region where the dynamics become truly nonlinear. Since the matching condition which the nonlinear dynamics imposes on the linear leading edge dynamics is still the same in this case [7,16], the above results even apply to incoherent fronts. How-ever, the phase relaxation applies in that case only to the coherent dynamics in the leading edge.
If the pattern forming front is coherent, the results apply throughout the whole front region. More precisely, we call a front coherent if the asymptotic front solution is time periodic in the co-moving frame ξ = x− v∗t, i.e. if there is
some period T such that
Φ(ξ, t + T ) = Φ(ξ, t), where φ(ξ, t)t→∞= Φ(ξ, t). (47)
The dynamics of the leading edge actually determines this period to be
T = 2π/Ω∗, (48)
where Ω∗ is the frequency determined by the saddle point (1). This can be
easily read from Eq. (5) or from Eq. (35) and the knowledge that ψX(ξX, t)
becomes stationary for t→ ∞.
6 Of course, for uniformly translating fronts there is no phase, hence q∗ = 0 = Ω∗
Because of the temporal periodicity, we can generally write a coherent Φ in the whole spatial domain as a Fourier series
Φ(ξ, t) = X
n=0,±1,...
e−inΩ∗tΦn(ξ). (49)
In our analysis [6] of fronts which converge to a uniformly translating front so-lution, we explicitly showed that to order O(1/t2), the front shape relaxation
follows the velocity relaxation adiabatically. An extension of the analysis to coherent pattern forming fronts shows that a similar result holds for these. The reason is that when the front is converging to its asymptotic shape as 1/t, the temporal derivative terms in the dynamical equations only generate terms of order 1/t2 in the asymptotic expansion, while the terms coming from
the adiabatic variation of v(t) and Γ(t) generate terms of order 1/t and 1/t3/2.
In other words, to order 1/t3/2 the only temporal dependence comes in
para-metrically via v(t) and Γ(t). Thus, for long times, coherent pattern forming fronts relax to their asymptotic shape according to
φ(x, t)t≫1= Φv(t)(ξX, t) +O(t−2) with Φ(ξX, t)≈ Φv(t)(ξX, t + T (t)), (50)
where v(t) and Γ(t) are given by Eqs (45) and (46) above, and where T (t) is the instantaneous period 2π/(Ω∗+ ˙Γ(t)). In terms of the temporal Fourier
series this result can be written as φ(x, t)t≫1= X
n=0,±1,···
e−in(Ω∗t+Γ(t))Φnv(t)(ξX) +O(t−2) (51)
where the Φn
v are the Fourier transform functions of the coherent pattern
forming solutions7 with velocity v and frequency Ω∗ + ˙Γ. Thus the above
result expresses that the coherent front profiles follow this family of solutions adiabatically, and that their velocity and frequency shift ˙Γ is set completely by the dynamics in the leading edge.
7 Clearly, this result implies the existence of a two-parameter family of coherent
4 Numerical study of the relaxation behavior of fronts in the Swift-Hohenberg equation
We now illustrate the above analysis with numerical results obtained for the Swift-Hohenberg equation (10). This equation has often been used [17,18,19,20,21,22] as one of the simplest equations to illustrate the behavior of coherent pattern forming fronts. Collet and Eckmann were the first to prove that fronts propa-gating into the linearly unstable state φ = 0 are pulled; the analysis of section 2 applies too and therefore establishes this fact as well. In the simulations of this equation presented here, we study the approach of the fronts to these asymptotic pulled front solutions, starting from a Gaussian initial condition. Note in this regard that while the Swift-Hohenberg is often studied for small ε where the dynamics maps onto an amplitude expansion, our front convergence analysis applies generally. We will illustrate this by taking finite values of ε. Fig. 1 shows a φ-profile for ε = 0.5.
We first illustrate an important ingredient of our convergence analysis. As we argued above, in the co-moving frame ξ = x− v∗t the leading edge variable ψ
defined in (30) should asymptotically behave as ξ/(t3/2)e−ξ2
/(4Dt)[Cf. Eq. (38)].
To illustrate this for the Swift-Hohenberg equation, we show in Fig. 2 three snapshots of the leading edge variable t3/2ψ(x, t) = eˆ λ∗(x−v∗t)
φ(x, t) in a sim-ulation for ε = 0.5; according to our analysis, the envelope of this function should asymptotically behave as
(x− v∗t) e−(x−v∗t)2
/(4Dt), with 1
D ≡ Re
1
D. (52)
Our numerical results in Fig. 2 fully confirm this behavior.
To test our convergence results, we have to extract the velocity v(t) and fre-quency Ω∗+ ˙Γ(t) from our numerical data. Because of the oscillating
charac-ter of the fronts, this is nontrivial in principle. We will do it in a pragmatic way, replacing differentials by finite difference approximants: In our simula-tion, we keep track of the local maxima of φ(x, t) and from these determine the positions Xn and times tn at which the foremost maximum n reaches
a predetermined fixed “level” ℓ. From this we calculate the finite difference approximants vℓ(tn) = Xn− Xn−1 tn− tn−1 , Ωℓ(tn) = 2π tn− tn−1 . (53)
0 200 400 600 800
x
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 ψ t 3/2 t=50 t=100 t=200Fig. 2. Three snapshots of the function t3/2ψ obtained from our simulations of theˆ Swift-Hohenberg equation for ε = 0.5. The results confirm the asymptotic behavior (52). Note in particular the diffusive broadening of the pattern: the one at time t = 200 is twice as wide as the one at time t = 50.
For testing the convergence up to terms of O(1/t3/2), the discretization error
is therefore irrelevant.
In Fig. 3 we show two plots of the velocity relaxation data for two different values of ε, namely ε = 0.5 and ε = 5. The various lines indicate the velocity extracted for different levels ℓ. To probe the predicted behavior in detail, we have plotted vℓ(t)−v∗−c1/t versus t−3/2. According to our prediction (45) this
velocity difference should asymptotically approach 0 along the dashed lines. Similar plots for the frequency relaxation, obtained from the same runs, are shown in Fig. 4. Clearly, all our numerical results are in full agreement with the predicted behavior.
We finally study the convergence of the shape of the profile to its asymptotic form. In principle, the information is contained in the expression (51) above, but to make it explicit one would have to know all functions Φn
v. Since our
goal here is simply to check that the shape relaxation follows the velocity and phase relaxation adiabatically, we circumvent this problem as follows. We construct an effective (real) envelope A(ξX, t) of the front profile8 in the
co-moving frame by tracking the positions of the maxima of φ(x, t) during one effective period 2π/(Ω∗ + ˙Γ(t)). In doing so, ξ
X is determined by requiring
that A(ξX = 0, t) = const. where the constant is chosen so that the level of
the effective envelope at this point is about half of its asymptotic value. The implication of (51) now is that the convergence of the effective envelope A(ξ, t) determined this way should, up to terms of O(1/t2), adiabatically follow the 8 Note that this real envelope A differs from the complex amplitude A of the
(a) 0 0.00025 0.0005 0.00075 0.001 0.00125 t-3/2 0 0.0025 0.005 0.0075 0.01 0.0125 0.015 v(t)-v * -c 1 /t num. data c3/2t -3/2 (b) 0 0.005 0.01 0.015 t-3/2 0 0.01 0.02 0.03 v(t)-v * -c 1 t -1 num.data c3/2t -3/2
Fig. 3. Velocity difference vℓ(t)− v∗− c1/t as a function of t−3/2 for ε = 0.5 (panel
a) and ε = 5 (panel b). The various lines denote, from top to bottom, the levels ℓ = 0.0001√ε, 0.001√ε, 0.01√ε, 0.05√ε, 0.2√ε, 0.3√ε and 0.5√ε. The dashed line is the asymptotic slope according to the exact expression (45).
(a) 0 0.0005 0.001 0.0015 t-3/2 0 0.005 0.01 0.015 -Ω (t)+ Ω * +d 1 t -1 (b) 0 0.01 0.02 0.03 t-3/2 0 0.01 0.02 0.03 0.04 0.05 -Ω (t)+ Ω * +d 1 t -1
Fig. 4. As Fig. 3, but now for the frequency relaxation Ω(t) = Ω∗+ ˙Γ(t).
velocity and shape relaxation:
A(ξX, t) = Av(t), ˙Γ(t)(ξX) +O(1/t2), (54) so that A(ξX, t)− A(ξX, t′) = δAv, ˙Γ(ξX) δv [v(t)− v(t ′)] +δAv, ˙Γ(ξX) δ ˙Γ [ ˙Γ(t)− ˙Γ(t ′)] +O(1/t2). (55)
As in the discretization (53), the averaging over one period only affects the terms of O(1/t2) in this expression.
Fig. 5 shows the effective envelope A(ξX, t) for the front from Fig. 1. The
figure confirms that even for this value, where the pattern behind the front is rapidly oscillating, the effective envelope can be obtained accurately and is smooth.
-40 -20 0 20 40 ξ 0.0 0.2 0.4 0.6 0.8 1.0 front envelope A
Fig. 5. The front envelope A(ξX, t) for ε = 0.5 obtained as described in the text. In
this case, t = 160, and the front shape is obtained by averaging over one period that lasts about ∆t = 2. Note the different horizontal scale in comparison with Fig. 1.
(a) -40 -20 0 20 40 ξ -0.015 -0.01 -0.005 0 0.005 0.01 0.015 A(t)-A(180) t=20 t=40 t=60 t=80 t=160 t=140 t=120 t=100 (b) -40 -20 0 20 40 ξ -0.015 -0.01 -0.005 0 0.005 0.01 0.015 A(t)-A(180)/(1/t-1/180)
Fig. 6. (a) The convergence of the effective envelope difference A(ξX, t)−A(ξX, 180),
as obtained from the numerical solutions illustrated in Fig. 5. (b) The ratio (56) as obtained from the data shown in panel (a). The figure confirms that this ratio converges to a time-independent function, in agreement with our predictions.
profile. Panel (a) shows the difference A(ξX, t)−A(ξX, 180), while in panel (b)
we plot the ratio
A(ξX, t)− A(ξX, 180)
1/t− 1/180 , (56)
5 Conclusion
In this paper we have presented two types of results. First of all, we have intro-duced a simple line of analysis which allows us to prove for certain classes of equations which include the Swift-Hohenberg equation, the Extended Fisher-Kolmogorov equation and the cubic Complex Ginzburg Landau equation that fronts are pulled. The method works for real or complex equations and fields and is not restricted to nonlinearities like N (A) A = |A|2nA with integer n,
but also can treat nonlinearities that depend, e.g., on ∂xA. Important is that
the over-all sign of Re N can be determined.
A Derivation of Eqs. (45) and (46)
The derivation follows essentially the lines of [6], except that z is now a com-plex rather than a real variable, and that there are additional terms due to q∗ and ˙Γ. The task is to solve (43) with the ansatz (44) and with boundary con-ditions (41) and (42). Actually, the analysis of the nonlinear region for finite t contributes additional terms to (42) which will play a role in the calculation of the subleading terms. The boundary conditions for ψX become
ψX(ξX, t) = αξX + β + f1(ξX) t + O f3/2(ξX) t3/2 ! , (A.1) ψX(ξX, t) ξ2 X/(4Dt)≫1 −→ 0. (A.2)
Insertion into the ansatz (44) implies for the function G(z, t) that
G(z, t) =√th2α√Dz + Oz3/2 i+hβ + O (z)i+O ( √ z) √ t + O 1 t , (A.3) lim z→∞e −z G(z, t) = 0. (A.4)
These boundary conditions determine a unique solution for the functions g1/2(z) and g0(z) and the constants c1, d1, c3/2 and d3/2 in ˙X and ˙Γ, as we will
derive below.
Inserting (44) into (43), we see that the dominant terms are of order t1/2.
Upon collecting these, we get
" z d 2 dz2 + 1 2− z d dz − 1 − λ ∗c 1 + i(d1+ q∗c1) # g−1/2 = 0. (A.5)
This homogeneous equation is an example of Kummer’s equation [28] ˆ T [a, b]g ≡ " z d 2 dz2 + (b− z) d dz − a # g = 0, (A.6)
whose general solution is a superposition of the two confluent hypergeometric functions
These functions are defined through the series M(a, b, z) = ∞ X n=0 (a)nzn (b)nn! , (A.8) where (a)n= a(a + 1) . . . (a + n− 1) = Γ(a + n)
Γ(a) , (a)0 = 1. (A.9)
The asymptotic large-z behavior of the functions M for positive b is
M(a, b, z)z→∞≃ Γ(b)
Γ(a) za−b ez for a6= 0, −1, −2, −3, · · · , (a)|a|z|a|
(b)|a|(|a|)! for a = 0,−1, −2, −3, · · · ,
(A.10)
Let us return to Eq. (A.5) for g−1/2(z). The boundary condition (A.3) implies
g−1/2(z) = 2α√Dz + O(z3/2). (A.11)
Since M(a, b, z = 0) = 1, a contribution of the solution M(a, b, z) is excluded through (A.11), and g−1/2(z) has to be proportional to z1−bM(1+a−b, 2−b, z).
With boundary condition (A.3), we therefore get g−1/2(z) = 2α√Dz M 3 2 + λ ∗c 1− i(d1+ q∗c1), 3 2, z . (A.12)
Furthermore, (A.10) shows that the Kummer functions M(a, b, z) diverge as ez when the coefficient a is not zero or a negative integer, while they are simple
polynomials when a is zero or a negative integer since then the coefficients (a)n
vanish for n≥ 1 − a. An exponential divergence of g is not allowed according to the second boundary condition (A.4); this fixes
1 + a− b = 3 2 + λ
∗c
1− i(d1 + q∗c1) = 0,−1, −2, . . . . (A.13)
For a detailed discussion of the solutions with 1 + a− b = −1, −2, . . ., we refer to [6]: essentially, these solutions are dynamically not relevant since they will always be overrun by the solution with 1 + a− b = 0. As both c1 and d1 are
real, (A.13) with 1 + a− b = 0 implies c1 =−
3
with the corresponding solution
g−1/2(z) = 2α√Dz. (A.15)
The terms of order t0 obtained by subsituting (44) into (43) are
ˆ T h12 + λ∗c1− i(d1+ q∗c1),12 i g0(z) = " −ik∗c 3/2− c1 √ z √ D(∂z − 1) − id3/2 # g−1/2(z) −D3 √ z D32 3 2(∂z− 1) 2 + z (∂z− 1)3 g−1/2(z) (A.16) −w √ z √ D 1 2− z(∂z− 1) − 1 − ik ∗c 1− id1 (∂z− 1) g−1/2(z).
The function g−1/2(z) on the right hand side of (A.16) is known from (A.15); likewise c1 and d1 are known from (A.14). Substitution of these results gives
the following inhomogeneous equation for g0(z)
ˆ T [−1,1 2] g0(z) = 2α h c3/2λ∗− i(d3/2+ q∗c3/2) i√ Dz + 3α 2λ∗ (1− 2z) +2α D3 D z2 − 3z + 3 4 + 2α w z2− 3z + 3 4 . (A.17)
The general solution of this inhomogeneous equation is a particular solu-tion plus the sum of two independent solusolu-tions of the homogeneous equasolu-tion
ˆ T [−1,1
2]g0(z) = 0. The latter can again be written in terms of Kummer
func-tions. It is easy to find particular solutions which reproduce most of the terms on the right by noting that
ˆ T [−1,1 2] √ z =12√z, T [ˆ−1,1 2]1 = 1, ˆ T [−1,1 2]z = 1 2, T [ˆ −1, 1 2]z 2 =−z2 + 3z. (A.18)
With these terms, we can generate all the terms on the right hand side of (A.16), except for the term linear in z. We can generate this term by noting that the function
is proportional to a truncated Kummer series M(1,1
2, z) (see below) and solves
ˆ T [−1,1 2]FN(z) = zN −1 1 2 N −1 (N− 1) , hence ˆT [−1,1 2]F2(z) = 2z . (A.20)
Using all the results (A.7), (A.18) and (A.20), we can write the general solution of (A.17) as g0(z) = k0(1− 2z) + l0 √ zM −1 2, 3 2, z +4αhc3/2λ∗− i(d3/2+ q∗c3/2) i√ Dz (A.21) +3α 2λ∗ h 1− F2(z) i − 2α D 3 D + w z2− 3 4 . where we used the fact that M(−1,1
2, z) = 1− 2z. The parameters k0, l0, c3/2
and d3/2are again determined by the boundary conditions. First, the boundary
condition (A.3) implies for g0 that g0(z) = β +O(z). This gives with (A.21)
β +O(z) = k0+ 3α 2 1 λ∗ + D3 D + w +h4αc3/2λ∗− i(d3/2+ q∗c3/2) √ D + l0 i√ z +· · · . (A.22) The first term on the right determines the coefficient k0 in terms of α, β and
the other parameters, but this term is not needed in the sequel. The condition that the prefactor of the √z term on the right vanishes gives
c3/2λ∗ − i(d3/2+ q∗c3/2)
√
D + l0
4α = 0. (A.23)
Second, the boundary condition (A.4) imposes also for g0(z), that the function
does not diverge exponentially for large z. There are two terms in (A.21) which diverge exponentially: the Kummer function M, whose asymptotic behavior is given in (A.10), and the function F2(z). It is easy to see that for large z, we
have z2d2F2(z) dz2 ≃ M(1, 1 2, z) =⇒ F2(z)≃ √ πz−3/2ez . (A.24)
Therefore the requirement that the two exponentially divergent terms in g0(z)
cancel each other, translates into l0
4α +
3√π
Upon eliminating l0/α from equations (A.23) and (A.25) we simply get c3/2= 3√π 2(λ∗)2Re 1 √ D, d3/2 =− 3√π 2λ∗ Im 1 √ D − q ∗c 3/2. (A.26)
The second contribution to d3/2is just the contribution to the phase relaxation
which is induced by the relaxation of v(t). Upon substitution of these results in the expansions (32) for X(t) and (34) for Γ(t) we get the results (45) and (46).
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